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## The Porous Medium Equation: Mathematical Theory

*Juan Luis Vazquez*

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198569039
- eISBN:
- 9780191717468
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198569039.001.0001
- Subject:
- Mathematics, Mathematical Physics

The heat equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it ... More

## Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type

*Juan Luis Vázquez*

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199202973
- eISBN:
- 9780191707919
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199202973.001.0001
- Subject:
- Mathematics, Applied Mathematics

This book is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as ... More

## INTRODUCTION

*Juan Luis Vázquez*

### in The Porous Medium Equation: Mathematical Theory

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198569039
- eISBN:
- 9780191717468
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198569039.003.0001
- Subject:
- Mathematics, Mathematical Physics

This chapter introduces the mathematical theory of the nonlinear heat equation referred to as the porous medium equation (PME). This equation is one of the simplest examples of a nonlinear evolution ... More

## PROPAGATION PROPERTIES

*Juan Luis Vázquez*

### in The Porous Medium Equation: Mathematical Theory

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198569039
- eISBN:
- 9780191717468
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198569039.003.0014
- Subject:
- Mathematics, Mathematical Physics

This chapter introduces a study of the properties of the support and the free boundary of the solution in a several dimensional setting. It discusses in detail the property of finite propagation and ... More

## Lower bounds, contractivity, error estimates, and continuity

*Juan Luis Vázquez*

### in Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199202973
- eISBN:
- 9780191707919
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199202973.003.0005
- Subject:
- Mathematics, Applied Mathematics

This chapter considers a number of bounds that complement the study so far, which was centred on upper bounds. The first section concerns the existence of lower bounds for non-negative solutions. ... More

## EFFECT OF FLUCTUATIONS ON THERMOELECTRICITY AND HEAT TRANSPORT

*Anatoly Larkin and Andrei Varlamov*

### in Theory of Fluctuations in Superconductors

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198528159
- eISBN:
- 9780191713521
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528159.003.0010
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter introduces a phenomenological definition of the heat current. An explicit expression for the heat current operator in interacting electron system is derived in the framework of ... More

## Parabolic Equations

*Rüdiger Verfürth*

### in A Posteriori Error Estimation Techniques for Finite Element Methods

- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199679423
- eISBN:
- 9780191758485
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199679423.003.0006
- Subject:
- Mathematics, Applied Mathematics, Numerical Analysis

We derive a posteriori error estimates for space–time finite element and certain finite volume discretisations of linear and nonlinear parabolic equations of second order. The starting point of our ... More

## Degenerate Diffusion Operators Arising in Population Biology (AM-185)

*Charles L. Epstein and Rafe Mazzeo*

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.001.0001
- Subject:
- Mathematics, Probability / Statistics

This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as ... More

## Two Contrasting Characters: Fourier and Herapath

*Jennifer Coopersmith*

### in Energy, the Subtle Concept: The discovery of Feynman's blocks from Leibniz to Einstein

- Published in print:
- 2015
- Published Online:
- August 2015
- ISBN:
- 9780198716747
- eISBN:
- 9780191800955
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198716747.003.0011
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, History of Physics

Herapath was an eccentric Englishman who developed a kinetic theory of gases. He was the first to estimate the typical speed of a gas molecule. His paper was rejected by Humphry Davy, and Herapath ... More

## The Resolvent Operator

*Charles L. Epstein and Rafe1 Mazzeo*

### in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.003.0011
- Subject:
- Mathematics, Probability / Statistics

This chapter describes the construction of a resolvent operator using the Laplace transform of a parametrix for the heat kernel and a perturbative argument. In the equation (μ-L) R(μ) f = f, R(μ) ... More

## What Is a Spectral Model?

*T. N. Krishnamurti, H. S. Bedi, and V. M. Hardiker*

### in An Introduction to Global Spectral Modeling

- Published in print:
- 1998
- Published Online:
- November 2020
- ISBN:
- 9780195094732
- eISBN:
- 9780197560761
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195094732.003.0006
- Subject:
- Earth Sciences and Geography, Meteorology and Climatology

If one takes a closed system of the basic meteorological equations and introduces within this system a finite expansion of the dependent variables using functions such as double Fourier or ... More

## Maximum Principles and Uniqueness Theorems

*Charles L. Epstein and Rafe1 Mazzeo*

### in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.003.0003
- Subject:
- Mathematics, Probability / Statistics

This chapter proves maximum principles for two parabolic and elliptic equations from which the uniqueness results follow easily. It also considers the main consequences of the maximum principle, both ... More

## The Model Solution Operators

*Charles L. Epstein and Rafe1 Mazzeo*

### in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.003.0004
- Subject:
- Mathematics, Probability / Statistics

This chapter introduces the model problems and the solution operator for the associated heat equations. These operators give a good approximation for the behavior of the heat kernel in neighborhoods ... More

## Holder Estimates for General Models

*Charles L. Epstein and Rafe1 Mazzeo*

### in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.003.0009
- Subject:
- Mathematics, Probability / Statistics

This chapter presents the Hölder estimates for general model problems. It first estimates solutions to heat equations for both the homogeneous Cauchy problem and the inhomogeneous problem, obtaining ... More

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